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Computable Functions

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A. Shen; N. K. Vereshchagin

In 1936, before the development of modern computers, Alan Turing proposed the
concept of a machine that would embody the interaction of mind, machine, and
logical instruction. The idea of a “universal machine” inspired the
notion of programs stored in a computer's memory. Nowadays, the study of
computable functions is a core topic taught to mathematics and computer science
undergraduates.

Based on the lectures for undergraduates at Moscow State University, this
book presents a lively and concise introduction to the central facts and basic
notions of the general theory of computation. It begins with the definition of
a computable function and an algorithm and discusses decidability,
enumerability, universal functions, numberings and their properties,
\(m\)-completeness, the fixed point theorem, arithmetical hierarchy,
oracle computations, and degrees of unsolvability. The authors complement the main
text with over 150 problems. They also cover specific computational models,
such as Turing machines and recursive functions.

The intended audience includes undergraduate students majoring in
mathematics or computer science, and all mathematicians and computer scientists
who would like to learn basics of the general theory of computation. The book
is also an ideal reference source for designing a course.

Readership

Undergraduates, graduate students, research mathematicians, and
computer scientists and programmers interested in the general theory of
computation.

Reviews & Endorsements

Material is a presented quite clearly and with a minimum of fuss
… an excellent text for a first course in computability.

In 1936, before the development of modern computers, Alan Turing proposed the
concept of a machine that would embody the interaction of mind, machine, and
logical instruction. The idea of a “universal machine” inspired the
notion of programs stored in a computer's memory. Nowadays, the study of
computable functions is a core topic taught to mathematics and computer science
undergraduates.

Based on the lectures for undergraduates at Moscow State University, this
book presents a lively and concise introduction to the central facts and basic
notions of the general theory of computation. It begins with the definition of
a computable function and an algorithm and discusses decidability,
enumerability, universal functions, numberings and their properties,
\(m\)-completeness, the fixed point theorem, arithmetical hierarchy,
oracle computations, and degrees of unsolvability. The authors complement the main
text with over 150 problems. They also cover specific computational models,
such as Turing machines and recursive functions.

The intended audience includes undergraduate students majoring in
mathematics or computer science, and all mathematicians and computer scientists
who would like to learn basics of the general theory of computation. The book
is also an ideal reference source for designing a course.